We study spin-transfer-torque-driven magnetization dynamics of a perpendicular magnetic tunnel junction nanopillar. Through the combined investigations on spin-torque ferromagnetic resonance and microwave spectroscopy, it is found that the free layer (FL) and the weak pinned reference layer (RL) exhibit distinct dynamic behaviors. First, frequency vs field dispersion for the FL and RL shows an opposite trend as the field varies. Second, the FL can support a single spin-wave (SW) mode for both parallel and antiparallel configurations, while the RL exhibits spin-wave excitation only for the antiparallel state. Those two SW modes coexist at the antiparallel state, and their oscillation frequency exhibits a crossover phenomenon with increasing the external magnetic field, which could be helpful in the mutual synchronization of auto-oscillations for SW-based neuromorphic computing.